2016 AMC 10B Problem 18

Question

In how many ways can

345

be written as the sum of an increasing sequence of two or more consecutive positive integers?

\ 1

\ 3

\ 5

\ 6

\ 7

If the length is

s

, then

s(2x+s-1)=690

Check factor lengths that give positive integer

x

Suppose the sequence has length

s

and first term

x

. Then

345=s\left(x+\frac{s-1}{2}\right),

or

s(2x+s-1)=690.

Thus

s

must be a divisor of

690

, with

x=\frac{\frac{690}{s}-s+1}{2}

a positive integer. Checking the possible divisor lengths gives

s=2,3,5,6,10,15,23.

These are the only lengths that keep

x

positive and integral.

Therefore there are

7

representations.

Thus, the correct answer is E.

Problem 18 in other years